Question

In Problems 55-58 use the given Laplace transform and the result in Problem 54 to find the indicated Laplace transform. Assume that a and k are positive constants.

56.$\mathcal{L}\mathbf{\left\{}\mathbf{sint}\mathbf{\right\}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{;}\mathbf{}\mathbf{}\mathbf{}\mathcal{L}\mathbf{\left\{}\mathbf{sinkt}\mathbf{\right\}}$

Short answer

$\mathcal{L}\left\{\mathrm{sinkt}\right\}=\frac{\mathrm{k}}{{\mathrm{s}}^2+{\mathrm{k}}^2}$

Step-by-step solution

Definition of Laplace Transform

Let f be a function defined for $\mathbf{t}\mathbf{⩾}\mathbf{0}$. Then the integral

$\mathcal{L}\mathbf{\left\{}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right\}}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{xt}}\mathbf{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{dt}$

is said to be the Laplace transform of f, provided that the integral converges.

Use Laplace Transform

$\mathrm{F}\left(\mathrm{s}\right)=\mathcal{L}\left\{\mathrm{f}\right(\mathrm{t}\left)\right\}=\mathcal{L}\left\{\mathrm{sint}\right\}=\frac{1}{{\mathrm{s}}^2+1}$

By definition of the change of scale theorem.

$\mathcal{L}\left\{\mathrm{f}\right(\mathrm{at}\left)\right\}=\frac{1}{\mathrm{a}}\mathrm{F}\left(\frac{\mathrm{s}}{\mathrm{a}}\right)$

Thus,

$\mathcal{L}\left\{\mathrm{sinkt}\right\}\stackrel{\left(2\right)}{\text{=}}\frac{1}{\mathrm{k}}\mathrm{F}\left(\frac{\mathrm{s}}{\mathrm{k}}\right)$

$\stackrel{\left(1\right)}{=}\frac{1}{\mathrm{k}}\left(\frac{1}{{\left(\frac{\mathrm{s}}{\mathrm{k}}\right)}^2+1}\right)$

$=\frac{1}{\mathrm{k}}\left(\frac{{\mathrm{k}}^2}{{\mathrm{s}}^2+{\mathrm{k}}^2}\right)$

Therefore,$\mathcal{L}\left\{\mathrm{sinkt}\right\}=\frac{\mathrm{k}}{{\mathrm{s}}^2+{\mathrm{k}}^2}$